
\((\alpha ,\beta )\) -transformations.
The \((\alpha ,\beta )\) -transformation \(T_{\alpha ,\beta }\colon [0,1]\rightarrow [0,1]\)  with \(\beta >1\)  and
\(0\le \alpha <1\)  was introduced by Parry ([1]}) and defined by
\(T_{\alpha ,\beta }(x)=\left\lbrace \begin{array}{ll}\beta x+\alpha \ (\text{mod}\ 1) & (x\ne 1),\\\displaystyle \lim _{y\rightarrow 1-0}(\beta y+\alpha \ (\text{mod}\ 1)) & (x=1).\end{array}\right.\) 

Generalized \(\beta \) -transformations.
Let \(\beta >1\)  and \(k\)  be a smallest integer
not less than \(\beta \) ,
fix \(E=(E_1,\ldots ,E_k)\in \lbrace +1,-1\rbrace ^k\) ,
and consider \(k\)  intervals
\(I_1:=[0,1/\beta ),\ I_2:=[1/\beta ,2/\beta ),\ldots ,I_k:=[k-1/\beta ,1].\) 
The generalized \(\beta \) -transformation
\(T_{\beta ,E}\colon [0,1]\rightarrow [0,1]\)  was
introduced by Góra ([2]}) and defined by
\(T_{\beta ,E}(x)=\left\lbrace \begin{array}{ll}\beta x-i+1 & (x\in I_i,\ E(i)=+1),\\-\beta x+i & (x\in I_i,\ E(i)=-1).\end{array}\right.\) 
If \(E=(-1,\ldots ,-1)\) , then
we call \(T_{\beta ,E}\)  a \((-\beta )\) -transformation.

